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Let $X \subset \mathbb{CP}^N$ a $k$-dimensional projective variety cut out by polynomials of degree $\leq d$ and $f_0,\cdots,f_N$ be homogeneous polynomials of degree $d$ without commom factors with the property that $Z_k = Z(f_0) \cap \cdots\cap Z(f_k)$ is of codimension $k+1$ and do not intersect $X$.

I want to proove that there exists a homogeneous polynomial $h \in I(X)$ of degree $d$ such that the hypersurface $V=Z(h)$ do not contain any component of $Z_k$ (i.e., $\text{codim} Z_k \cap X = k+2$)

Naively speaking the variety $Z_k$ has degree $d^{k+1}$ and it could not be contained in something of degree $d$, but it may happen that $Z_k$ has a component of degree $<d$ and this component could be contained in $V$, that is why we should choose $h$ conveniently.

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